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Siteswap
Siteswap (also called Quantum Juggling or Cambridge notation) is the most common system of juggling notation. It was independently invented by Paul Klimek in 1981, by Bruce Tiemann and Bengt Magnusson in 1985, and by Mike Day, Colin Wright, and Adam Chalcraft in 1985. Siteswap is a useful tool for communicating between jugglers, discovering new patterns, and finding transitions between different patterns. Many Juggling Animators use siteswap notation as their pattern input. Siteswap notation can be used to describe synchronous or asynchronous patterns for any number of objects and any number of people, with various throw heights, crossing and non-crossing throws, pauses in the pattern, passing objects directly from one hand to another, multiplex throws and squeeze catches, and transitions between sync and async patterns. Siteswap does not specify the positions of the throws (like backcrosses or under the leg throws), the positions of the catches (like penguins or blind catches), the paths the props take between being thrown and caught (like outside throws or bounced throws), how the props rotate in the air (like pancakes or helicopters), or things that are added to the juggling but aren't actually part of the pattern (like 360s or balances). 640px|leftRainbow siteswaps with programmable glow balls by Phillip Edwards 640px|left 3 rounds of db97531 by Lauge Benjaminsen Basic notation A siteswap pattern or "a siteswap" is a sequence of numbers. Each number represents a type of throw to be done at that time or "beat" in the sequence. A number in a siteswap generally represents the kind of throw that would be done in the basic pattern for that number of objects (juggled at the same speed as the siteswap). In an asynchronous siteswap, odd numbers represent throws that cross from one hand to the other, and even numbers represent throws that are thrown and caught with the same hand. Numbers greater than 9 are written as letters,so that "10" is "a" and "11" is "b" and so on. (This is not hexadecimal, it's just writing numbers as letters to avoid ambiguous double-digits; numbers beyond f continue as g, h, i... not 10, 11, 12...) Strings of values have been written separated as "6 4 5 1" and "10 8 6 4", but with alphanumeric digits the spaces are unnecessary, so siteswap strings are generally written without spaces, e.g. 6451 and a864. Basic (sometimes called "vanilla") siteswap also includes these assumptions: *All tosses are made to a strict beat so that every throw is on a beat. *Each hand throws in turn, right-left-right-left (called "asynchronous"). *Each toss and catch is of only one prop. (But see synchronous notation, multiplex notation, passing notation and diabolo siteswaps below.) A number in a siteswap sequence corresponds to the number of beats later an object will be thrown again after the throw that number represents. This usually corresponds to how high it was thrown (higher siteswap number = higher throw), so many people refer to the numbers as heights, but this is not technically correct; all that matters is the number of beats in the air, not how high it is thrown. For example, letting a ball bounce off the floor between throwing and catching it takes longer than a throw in the air at the same height, and so the bounced throw can be a higher siteswap value while being a lower throw. Also, the throw heights (for normal throws in the air) are not proportional to the siteswap numbers; a 6 is more than twice the height of a 3 (see the height formula below). There are three numbers with special meanings: a "0" is a pause with an empty hand, a "1" is a quick pass straight across to the other hand (also called a hand off, a feed, a zip, or a vamp), and a "2" is a pause with an object held in the hand. These are all things that can be done continually with their respective numbers of objects. It is assumed that the sequence shown as defining a pattern repeats indefinitely, so that 5 ball cascade, which is tossed as ...55555..., is notated as only "5". By this convention too, the notation 12345 would be "...1234512345..." allowing a variety of starting points for the notation. For this reason, "12345" is the same pattern as "23451" or "34512" and so on. If no number is shifted in the sequence, and the whole sequence is shown, the pattern shown is the same. The conventional ordering of a siteswap string is by that needed to start from holding all the objects, which is the same as directly connecting from the basic pattern, so "345" is written as 534. Which toss is by a left or right hand is not specified, only the regular designation for each hand next across the code string. A pattern can start with either hand, and then the throws alternate between hands: right-left-right-left-right-left... or left-right-left-right-left-right... The average of the numbers in a siteswap is the number of objects used in the pattern. All siteswaps must have a whole number average, but not every sequence with a whole number average is a valid siteswap. To be jugglable, a sequence cannot have a number followed 1 beat later by 1 less than that number, or a number followed 2 beats later by 2 less than that number, a number followed 3 beats later by 3 less than that number, etc. Every invalid sequence with a whole number average can be rearranged at least one way to get a valid siteswap. The actual height of a throw in a siteswap is equal to g(st - 2dt)^2 / 8, where g = acceleration due to gravity (about 32.17 feet per second per second, or 9.8 metres per second per second), s = the siteswap number, t = the amount of time between throws, and d = the dwell ratio. The "period" of a siteswap is the number of beats in the siteswap before it repeats, e.g. 97531 is a period 5 siteswap, repeating the same sequence of throws every 5 beats. Adding the period of a siteswap to one of the numbers in the siteswap makes a valid siteswap for one more object, and subtracting the period of a siteswap from one of the numbers in the siteswap makes a valid siteswap for one less object. Valid siteswaps for one more or one less object can also be made by adding 1 to or subtracting 1 from each of the numbers in a siteswap. Patterns that can be entered directly from the basic pattern (such as 744) are called ground state siteswaps; patterns that require special transition throws to enter starting from the basic pattern (such as 771) are called excited state siteswaps. 640px|left 5 ball siteswaps by Tsubasa Murakami Synchronous notation Synchronous siteswap notation is very much like vanilla siteswap notation above, but with two differences: (1) we group together right- and left-hand throws made simultaneously with parentheses, separating the two throws with a comma, and (2) we now need to specify a destination hand for each throw, so put an "x" after any throw that crosses over into the other hand. (A throw without the "x" is assumed to be directed to the same hand that threw it. An "x" after an odd number makes it a non-crossing throw, and is only used in transitions between sync and async patterns.) A "2" is still a hold, but a "2x" is similar to the "1" in async notation: a quick pass from one hand to the other (something that can be done continually with 2 balls in a synchronous pattern). A "0" is still an empty hand, and a "0x" is normally not allowed (see "Impossible Siteswaps" below). All throw numbers in a sync pattern must be even. An asterisk (read as "star") at the end of a sync pattern means that each time the sequence is repeated, the roles of the hands are reversed, e.g. (4,2x)* is short for (4,2x)(2x,4). Multiplex notation A multiplex throw is written in siteswap notation as two or more numbers in brackets. 6 balls juggled in a 3 ball cascade (6 ball stacks) would be written as 33. Passing notation A "throw" in passing notation is divided into multiple parts, the throwing instructions for each juggler. The notation <3|3> describes 2 jugglers each doing a 3 ball cascade. The "|" symbol separates throws made simultaneously by different jugglers. Normally the throws within a <|> are either all right-hand throws or all left-hand throws, but not all patterns have both right hands throwing at the same time. Juggling Lab uses the notation before a pattern to indicate that one person throws with the right hand while the other person throws with the left hand. A "p" after a number represents a passed throw. If the throw without a "p" would go to your left hand, the throw goes to your partner's left hand (on your right), and vice versa. If there are more than two jugglers, a number can be used after the "p" to indicate which juggler you're passing to, with the convention that the leftmost juggler in the <|> is juggler #1, the next is juggler #2, and so on. So for example, <3p|3p> and <3p2|3p1> each describe the same passing pattern, namely 6-object ultimates. Diabolo siteswaps Siteswap notation can also be used for diabolo patterns. Since diabolos are juggled with only one manipulator (the string) instead of two hands, the numbers have different meanings. A number still represents the kind of throw that would be done in the basic pattern for that number (of diabolos), but the only difference in the throws is the height, instead of having crossing and non-crossing throws. A "0" in diabolo siteswap is a pause with an empty string, and a "1" is a sun. 640px|left 4 diabolo siteswaps by Guillaume Karpowicz Impossible siteswaps Siteswap notation can be used to denote patterns that are valid in theory but don't work in practice because they require teleportation or time travel. A "0x" in synchronous notation would be an object instantly relocating from one hand to the other at the moment it's thrown. (This is a crossing version of an empty hand beat (a 0); the object switches from one hand to the other, resulting in an empty hand. The object crosses instantly because a 0 is "throwing" an object so that it will be thrown again 0 beats later.) In the 1 ball pattern (2x,0x), a ball would continually be thrown horizontally from the right hand and caught in the left hand, but it would never be thrown back to the right hand; it would instantly jump back to the right hand just before each throw. In the 2 ball pattern (4x,0x)*, the hands alternate making crossing throws at 4 ball height, like in the slow 3 ball cascade (4x,2)*, but after a ball is caught in one hand, it instantly jumps back to the hand that threw it, just before that same hand throws it again. The 3 ball pattern (4,4)(4x,0x) is similar to the sync fountain with a hole (4,4)(4,0), but the ball that isn't part of a synchronous pair is thrown across from one hand to the other, and after it's caught and ready to be thrown again, it instantly goes back to the hand that threw it and gets thrown by that hand again. (6,0x)* is like the slow 3 ball cascade (6x,0)*, but the throws don't cross, and the balls instantly teleport to the other side of the pattern when they're thrown. (0x,0x) is essentially the same thing as (0,0). Any pattern with a 0x can be changed into a possible pattern by changing the crossing/non-crossing status of each number in the synchronous pair that has a 0x, and switching the two numbers in the pair: (2x,0x) becomes (0,2), (4x,0x)* becomes (0,4)*, (4,4)(4x,0x) becomes (4,4)(0,4), (6,0x)* becomes (0,6x)*, (6x,4)(0x,2x)* becomes (6x,4)(2,0)*. A 0x can also be interpreted as a quick pass from hand to hand similar to a 1 to make a jugglable pattern: (2x,0x), (4x,0x)*, (4,4)(4x,0x), (6,0x)*. A negative number in a siteswap represents a throw that goes backwards in time. The next time a ball thrown as a -1 (a crossing throw) will be thrown again is 1 beat before that throw was made. A -2 is thrown again with the same hand 2 beats before that throw, and a -3 is thrown again with the opposite hand 3 beats before that throw. A negative number throw would look like a catch, and a negative number catch would look like a throw. When a negative throw is caught to be thrown again as a positive throw, it looks like two balls are appearing in an empty hand and being thrown as a multiplex (or only one of the balls is thrown and the other stays in the hand). (Both are actually the same ball; one is the ball going forwards in time, and the other is the ball going backwards in time.) When a positive throw is caught to be thrown again as a negative throw, it looks like two balls are caught in a sqeeze catch (or a ball is caught in a hand that's already holding a ball), and then they disappear. Patterns with a mix of positive and negative throws look like they're done with more objects than there really are, because when an object changes from going forwards in time to going backwards in time, it goes back through the same period of time where it already exsisted during the previous throw, so there can two of the same object in different places at the same time. 640px|left Negative siteswaps with programmable glow balls by Phillip Edwards See also *Siteswap states *Sprung siteswaps *Stack notation *Beatmap *MMSTD External links *Siteswap.org *Siteswap article on Wikipedia *Juggling Lab